Britton was born in Waco, Texas. He attended Tomball High School. In 2006, his junior season, he was recognized as an Aflac All–American. Later that same year, he took part in the Junior Olympics operated by USA Baseball..
The Boston Red Sox selected Britton in the 23rd round of the 2007 MLB Draft. Britton signed a $700,000 signing bonus with the Red Sox. He began his professional career in 2008 with the Class A short-season Lowell Spinners. He appeared in eight games, making seven starts, and recorded an ERA of 4.28 over 33.2 innings.
In 2009, Britton split time between the Gulf Coast League Red Sox and Lowell; he combined to appear in 11.2 innings over seven games. He missed most of the season due to Tommy John surgery.
2010 saw Britton promoted to the Class A full-season Greenville Drive, where he started 21 games and compiled a 2.97 ERA.
Britton was promoted once again in 2011 to the Salem Red Sox. In Salem, Britton made 26 starts and posted a 1–13 record. His 13 losses were a league high and His 6.91 ERA was the worst for all Carolina League starters.
In November 2011, Britton was added to the Red Sox' 40-man roster to prevent other teams from selecting him in the Rule 5 Draft.
Britton split 2012 between Salem and the Double–A Portland Sea Dogs of the Eastern League.
Britton remained with Portland at the start of the 2013 season. He joined the Triple–A Pawtucket Red Sox on July 5, and was then called up to the major leagues on July 14. His MLB debut came on July 20 as he did not allow a run in one inning against the New York Yankees. On July 31, Britton tossed a pair of shutout innings against the Seattle Mariners to earn his first MLB win.
Britton was claimed off waivers by the Chicago Cubs on February 4, 2015. He was outrighted off the roster on April 3. During the 2015 season, Britton played for the Triple-A Iowa Cubs. He pitched in 28 games, including 11 as a starter, and compiled a 7–8 record with one save, a 5.08 ERA, and 45 strikeouts. An injury to his left shoulder caused Britton to spend some of late August and early September on the disabled list.
On December 11, 2015, Britton signed a minor-league contract with the Detroit Tigers, and was invited to spring training. In 37 appearances for the Toledo Mud Hens, Detroit's AAA affiliate, Britton posted a 4.57 ERA over 41.1 innings. On August 2, 2016, Britton was suspended 50 games for violating the terms of the Major League Baseball drug policy.
On March 14, 2017, Britton signed with the Bridgeport Bluefish of the Atlantic League of Professional Baseball for the 2017 season.
On March 5, 2013, he was pulled over driving 113 miles per hour (182 km/h) in a 45 miles per hour (72 km/h) zone at 4:14am. He allegedly handed police his debit card when asked for ID. He was charged with DUI. The incident happened hours before a scheduled spring training start with the Red Sox.